lecturereservoir operation

8/9/2019 LectureReservoir operation

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Water Resources Systems Planning and Management: Linear Programming and Applications: Reservoir

Operation and Reservoir Sizing using LP

D Nagesh Kumar, IISc, Bangalore

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M3L6

MODULE – 3 LECTURE NOTES – 6

RESERVOIR OPERATION AND RESERVOIR SIZING USING LP

INTRODUCTION

In the previous lectures, we discussed applications of LP in deciding the optimal irrigation

allocation and water quality management. In this lecture we will discuss about the

applications of LP in modeling reservoir operation and reservoir sizing.

RESERVOIR OPERATION

Reservoir operation policies are developed to enable the operator to take appropriate

decision. The reservoir operation policy indicates the amount of water to be released based on

the state of the reservoir, demands and the likely inflow to the reservoir. The release from a

single purpose reservoir can be done with the objective of maximizing the benefits. For

multi-purpose reservoirs, there is a need to optimally allocate the releases among purposes.

The simplest of the operation policies is the standard operation policy (SOP). According to

SOP, if the water available (storage, S t + inflow, I t ) at a particular period is less than the

demand Dt , then all the available water is released. If the available water is more than the

demand but less than demand + storage capacity K , then release is equal to the demand. If

after releasing the demands, there is no space for extra water, then the excess water is alsoreleased. This is shown graphically in figure 1.

Fig. 1 Standard Operating Policy

D

D

Release

D + K Available water =

Storage + Inflow

45

O

A

C

B


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Along OA: Release = water available; reservoir will be empty after release.

Along AB: Release = demand; excess water is stored in the reservoir (filling phase).

At A: Reservoir is empty after release.

At B: Reservoir is full after release.

Along BC: Release = demand + excess of availability over the capacity (spill)

The releases according to the SOP need not be optimum. The optimization of reservoir

operation is done often by linear programming (LP) and dynamic programming (DP). DP will

be explained in the next module.

Derivation of optimal operating policy using LP

Consider a reservoir of capacity K . The optimization problem is to determine the releases R t

that optimize an objective function satisfying all the constraints. The objective function can

be a function of storage volume or release. The typical constraints in a reservoir optimization

model include conservation of mass and other hydrological and hydraulic constraints,

minimum and maximum storage and release, hydropower and water requirements as well as

hydropower generation limitations.

Fig. 2 Single reservoir operation

Consider the objective of meeting the demands to the extent possible i.e., maximizing the

releases. The optimization model can be formulated as:

Maximizet

t R

Subject to

(i) Hydraulic constraints as defined by the reservoir continuity equation

St+1 = St + It – EVt - R t - Ot for all t

where Ot is the outflow. The constraints for outflow are

Inflow, I t

Evaporation, EV t

Storage, S t

Release

(irrigation+

water su l , Rt


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Ot = 0 if St + It – EVt - R t ≤ K

= K – [St + It – EVt - R t] if St + It – EVt - R t > K

(ii) Reservoir capacity

St ≤ K – K d for all t, where K d is the dead storage

or simply St ≤ K

St ≥ 0 for all t.

(iii) Target demand

R t ≤ Dt for all t.

R t ≥ 0 for all t.

Large LP problems can be solved very efficiently using LINGO - Language for INteractive

General Optimization, LINDO Systems Inc, USA

Example

Derive an optimal operating policy for a reservoir to meet a long-term objective. Single

reservoir operation with deterministic inflows. K = 400.

Table 1. Inflow, evaporation and demand values of the reservoir

t Inflows Evaporation Demand

1 90.7 10 71.5

2 450.6 8 140.5

3 380.4 8 140.5

4 153.2 8 80.6

5 120 6 30.6

6 55 6 240.6

7 29.06 5 241.7

8 24.27 6 190.5

9 30.87 6 98.1

10 15.9 8 0

11 12.8 9 0

12 15.9 10 0

Solution

Objective function Maximizet

t R

Subject to



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where Ot is the outflow

Ot = 0 if St + It – EVt - R t ≤ K

= K – [St + It – EVt - R t] if St + It – EVt - R t > K

St ≤ 400 ; St ≥ 0; R t ≤ Dt ; R t ≥ 0 for all t.

The problem is solved using LINGO and the solution is given in table 2.

Table2. LP solution

t St It Dt R t EVt St+1 Ot

1 17.6 90.7 71.5 71.5 10 26.8 0

2 26.8 450.6 140.5 140.5 8 328.9 0

3 328.9 380.4 140.5 140.5 8 400 160.8

4 400 153.2 80.6 80.6 8 400 64.6

5 400 120 30.6 30.6 6 400 83.4

6 400 55 240.6 240.6 6 208.4 0

7 208.4 29.06 241.7 232.21 5 0.25 0

8 0.25 24.27 190.5 18.27 6 0.25 0

9 0.25 30.87 98.1 25.12 6 0 0

10 0 15.9 0 0 8 7.9 0

11 7.9 12.8 0 0 9 11.7 0

12 11.7 15.9 0 0 10 17.6 0

The rule curve derived is shown in figure 3.

Fig. 3 Rule curve

The optimal operation of a multipurpose single and multiple reservoir systems are discussed

in module 5.


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RESERVOIR SIZING

In many situations, annual demand may be less than the total inflow to a particular site.

However, the time distribution of demand and inflows may not match, which in turn result in

surplus in some periods and deficit in some other periods. Hence, there is a need of storage

structure i.e., reservoir to store water in periods of excess flow and make it available when

there is a deficit. In order to enable regulation of the storage to best meet the specified

demands, the reservoir storage capacity should be enough. The problem of reservoir sizing

involves determination of the required storage capacity of the reservoir when inflows and

demands in a sequence of periods are given. Reservoir capacity can be determined using two

methods: Mass curve method and Sequent peak algorithm method.

Mass diagram method

It was developed by W. Rippl (1883). A mass curve is a plot of the cumulative flow volumes

as a function of time. Mass curve analysis is done using a graphical method called Ripple’s

method. It involves finding the maximum positive cumulative difference between a sequence

of pre-specified (desired) reservoir releases R t and known inflows Qt (as shown in figure 4).

One can visualize this as starting with a full reservoir, and going through a sequence of

simulations in which the inflows and releases are added and subtracted from that initial

storage volume value. Doing this over two cycles of the record of inflows will identify the

maximum deficit volume associated with those inflows and releases. This is the required

reservoir storage.

Fig. 4 Typical mass curve

Time, t

C u m u l a t i v e i n f l o w

Release rate


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Sequent Peak Algorithm

This algorithm computes the cumulative sum of differences between the inflows and

reservoir releases for all periods t over the time interval [0, T]. Let K t be the maximum total

storage requirement needed for periods 1 through period t and Rt be the required release in

period t, and Qt be the inflow in that period. Setting K 0 equal to 0, the procedure involves

calculating K t using equation below for upto twice the total length of record. Algebraically,

otherwise

positiveif K Q R K

t t t

t 0

1

The maximum of all K t is the required storage capacity for the specified releases Rt and

inflows, Qt .

Formulation of reservoir sizing using LP

Linear Programming can be used to obtain reservoir capacity more elegantly by considering

variable demands and evaporation rates. The optimization problem is

Minimize K a

where K a is the active storage capacity

Subject to

(i)

Hydraulic constraints as defined by the reservoir continuity equation



St ≤ K a for all t

ST+1 = St where T is the last period.

(iii)

Target demands

R t ≥ Dt for all t.

STORAGE YIELD

A complementary problem to reservoir capacity estimation can be done by maximizing the

yield. Firm yield is the constant (or largest) quantity of flow that can be released at all times.

It is the flow magnitude that is equaled or exceeded 100% of time for a historical sequence of

flows. Linear Programming can be used to maximize the yield, R (per period) from areservoir of given capacity, K . The optimization problem can be stated as:


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Maximize R

Subject to

(i) Storage continuity equation



St ≤ K a for all t

ST+1 = St where T is the last period.

BIBLIOGRAPHY / FURTHER READING:

1.

Dennis T.L. and L.B. Dennis, Microcomputer Models for Management Decision Making,West Publishing Company, 1993.

2. Loucks, D.P., J.R. Stedinger, and D.A. Haith, Water Resources Systems Planning and

Analysis, Prentice-Hall, N.J., 1981.

3. Mays, L.W. and K. Tung, Hydrosystems Engineering and Management , Water Resources

Publication, 2002.

4. Rao S.S., Engineering Optimization – Theory and Practice, Fourth Edition, John Wiley

and Sons, 2009.

5. Taha H.A., Operations Research – An Introduction, 8th edition, Pearson Education India,

2008.

6.

Vedula S., and P.P. Mujumdar, Water Resources Systems: Modelling Techniques and

Analysis, Tata McGraw Hill, New Delhi, 2005.

7. Rippl., W., The capacity of storage reservoirs for water supply, Proceedings of the

Institution of Civil Engineers, 71:270 – 278.

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